Nondegeneracy of bubble solutions to the Choquard equation in two dimension

TL;DR

This paper proves the nondegeneracy of bubble solutions to a two-dimensional Choquard equation with exponential nonlinearity, extending known results from Liouville and higher-dimensional cases using integral and harmonic analysis.

Contribution

It establishes the nondegeneracy of solutions for the 2D Choquard equation, a problem previously unresolved, by combining integral representation and spherical harmonic decomposition.

Findings

Proves nondegeneracy of bubble solutions in 2D Choquard equation.

Extends nondegeneracy results from Liouville and higher-dimensional cases.

Uses integral representation and spherical harmonic analysis.

Abstract

In this paper, we study the following Choquard equation with exponential nonlinearity \begin{equation*} -\Delta u=\left(\int_{\R^{2}}\frac{e^{u(y)}}{|x-y|^{\alpha}}dy\right)e^{u(x)},\quad \text{~in~}\R^{2}, \end{equation*} where $\alpha\in (0,2)$. Although the classification of solutions to this equation has been established recently, the nondegeneracy of its solutions remains open. Here, we prove the nondegeneracy by combining the integral representation of solutions with the spherical harmonic decomposition. The main result of this paper can be viewed as an extension of the nondegeneracy of solutions for both the planar Liouville equation and the higher-dimensional upper critical Choquard equation.